Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Percentage Accurate: 87.6% → 99.8%

Time: 6.2s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))

C

double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

Python

def code(x, y):
	return (x * ((x / y) + 1.0)) / (x + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * ((x / y) + 1.0)) / (x + 1.0);
end

Wolfram

code[x_, y_] := N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))

C

double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

Python

def code(x, y):
	return (x * ((x / y) + 1.0)) / (x + 1.0)

Julia

function code(x, y)
	return Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * ((x / y) + 1.0)) / (x + 1.0);
end

Wolfram

code[x_, y_] := N[(N[(x * N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1 + \frac{x}{y}}{1 + x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ (+ 1.0 (/ x y)) (+ 1.0 x))))

C

double code(double x, double y) {
	return x * ((1.0 + (x / y)) / (1.0 + x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * ((1.0d0 + (x / y)) / (1.0d0 + x))
end function

Java

public static double code(double x, double y) {
	return x * ((1.0 + (x / y)) / (1.0 + x));
}

Python

def code(x, y):
	return x * ((1.0 + (x / y)) / (1.0 + x))

Julia

function code(x, y)
	return Float64(x * Float64(Float64(1.0 + Float64(x / y)) / Float64(1.0 + x)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * ((1.0 + (x / y)) / (1.0 + x));
end

Wolfram

code[x_, y_] := N[(x * N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.0%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
2. Step-by-step derivation

   1. associate-/l* 99.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]
4. Step-by-step derivation

   1. clear-num 99.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{x + 1}{\frac{x}{y} + 1}}{x}}} \]

   2. associate-/r/ 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{\frac{x}{y} + 1}} \cdot x} \]

   3. clear-num 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y} + 1}{x + 1}} \cdot x \]

   4. +-commutative 99.9%

      \[\leadsto \frac{\color{blue}{1 + \frac{x}{y}}}{x + 1} \cdot x \]

5. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{1 + \frac{x}{y}}{x + 1} \cdot x} \]

6. Final simplification 99.9%

   \[\leadsto x \cdot \frac{1 + \frac{x}{y}}{1 + x} \]

Alternative 2: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x}\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 700000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ 1.0 x))))
   (if (<= x -1.6e+69)
     (/ x y)
     (if (<= x 2.1e-91)
       t_0
       (if (<= x 6.2e-51)
         (* x (/ x y))
         (if (<= x 700000000000.0) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -1.6e+69) {
		tmp = x / y;
	} else if (x <= 2.1e-91) {
		tmp = t_0;
	} else if (x <= 6.2e-51) {
		tmp = x * (x / y);
	} else if (x <= 700000000000.0) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 + x)
    if (x <= (-1.6d+69)) then
        tmp = x / y
    else if (x <= 2.1d-91) then
        tmp = t_0
    else if (x <= 6.2d-51) then
        tmp = x * (x / y)
    else if (x <= 700000000000.0d0) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -1.6e+69) {
		tmp = x / y;
	} else if (x <= 2.1e-91) {
		tmp = t_0;
	} else if (x <= 6.2e-51) {
		tmp = x * (x / y);
	} else if (x <= 700000000000.0) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (1.0 + x)
	tmp = 0
	if x <= -1.6e+69:
		tmp = x / y
	elif x <= 2.1e-91:
		tmp = t_0
	elif x <= 6.2e-51:
		tmp = x * (x / y)
	elif x <= 700000000000.0:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (x <= -1.6e+69)
		tmp = Float64(x / y);
	elseif (x <= 2.1e-91)
		tmp = t_0;
	elseif (x <= 6.2e-51)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 700000000000.0)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (1.0 + x);
	tmp = 0.0;
	if (x <= -1.6e+69)
		tmp = x / y;
	elseif (x <= 2.1e-91)
		tmp = t_0;
	elseif (x <= 6.2e-51)
		tmp = x * (x / y);
	elseif (x <= 700000000000.0)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e+69], N[(x / y), $MachinePrecision], If[LessEqual[x, 2.1e-91], t$95$0, If[LessEqual[x, 6.2e-51], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 700000000000.0], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.59999999999999992e69 or 7e11 < x

   1. Initial program 74.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around inf 79.7%

      \[\leadsto \frac{x}{\color{blue}{y}} \]

   if -1.59999999999999992e69 < x < 2.0999999999999999e-91 or 6.1999999999999995e-51 < x < 7e11

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in y around inf 80.6%

      \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

   if 2.0999999999999999e-91 < x < 6.1999999999999995e-51

   1. Initial program 99.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]

   2. Taylor expanded in x around inf 94.9%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{x + 1} \]
   3. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{x + 1} \]

      2. associate-*r/ 94.7%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   4. Simplified 94.7%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   5. Taylor expanded in x around 0 94.9%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \]
   6. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \]

      2. associate-*r/ 94.7%

         \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]

   7. Simplified 94.7%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 700000000000:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x}\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ 1.0 x))))
   (if (<= x -1.3e+67)
     (/ x y)
     (if (<= x 2.6e-91)
       t_0
       (if (<= x 1.35e-51) (/ x (/ y x)) (if (<= x 1e+15) t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -1.3e+67) {
		tmp = x / y;
	} else if (x <= 2.6e-91) {
		tmp = t_0;
	} else if (x <= 1.35e-51) {
		tmp = x / (y / x);
	} else if (x <= 1e+15) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 + x)
    if (x <= (-1.3d+67)) then
        tmp = x / y
    else if (x <= 2.6d-91) then
        tmp = t_0
    else if (x <= 1.35d-51) then
        tmp = x / (y / x)
    else if (x <= 1d+15) then
        tmp = t_0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -1.3e+67) {
		tmp = x / y;
	} else if (x <= 2.6e-91) {
		tmp = t_0;
	} else if (x <= 1.35e-51) {
		tmp = x / (y / x);
	} else if (x <= 1e+15) {
		tmp = t_0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (1.0 + x)
	tmp = 0
	if x <= -1.3e+67:
		tmp = x / y
	elif x <= 2.6e-91:
		tmp = t_0
	elif x <= 1.35e-51:
		tmp = x / (y / x)
	elif x <= 1e+15:
		tmp = t_0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (x <= -1.3e+67)
		tmp = Float64(x / y);
	elseif (x <= 2.6e-91)
		tmp = t_0;
	elseif (x <= 1.35e-51)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 1e+15)
		tmp = t_0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (1.0 + x);
	tmp = 0.0;
	if (x <= -1.3e+67)
		tmp = x / y;
	elseif (x <= 2.6e-91)
		tmp = t_0;
	elseif (x <= 1.35e-51)
		tmp = x / (y / x);
	elseif (x <= 1e+15)
		tmp = t_0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e+67], N[(x / y), $MachinePrecision], If[LessEqual[x, 2.6e-91], t$95$0, If[LessEqual[x, 1.35e-51], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+15], t$95$0, N[(x / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3e67 or 1e15 < x

   1. Initial program 74.6%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around inf 79.7%

      \[\leadsto \frac{x}{\color{blue}{y}} \]

   if -1.3e67 < x < 2.60000000000000014e-91 or 1.3499999999999999e-51 < x < 1e15

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in y around inf 80.6%

      \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

   if 2.60000000000000014e-91 < x < 1.3499999999999999e-51

   1. Initial program 99.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]

   2. Taylor expanded in x around inf 94.9%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{x + 1} \]
   3. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{x + 1} \]

      2. associate-*r/ 94.7%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   4. Simplified 94.7%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   5. Taylor expanded in x around 0 94.9%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \]
   6. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \]

      2. associate-*r/ 94.7%

         \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]

   7. Simplified 94.7%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]
   8. Step-by-step derivation

      1. clear-num 94.7%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{x}}} \]

      2. un-div-inv 95.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{x}}} \]

   9. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 10^{+15}:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x}{y}\\ t_1 := \frac{x}{1 + x}\\ \mathbf{if}\;x \leq -330000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 105000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ x y))) (t_1 (/ x (+ 1.0 x))))
   (if (<= x -330000.0)
     t_0
     (if (<= x 2.8e-91)
       t_1
       (if (<= x 1.65e-51) (/ x (/ y x)) (if (<= x 105000000.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = x / (1.0 + x);
	double tmp;
	if (x <= -330000.0) {
		tmp = t_0;
	} else if (x <= 2.8e-91) {
		tmp = t_1;
	} else if (x <= 1.65e-51) {
		tmp = x / (y / x);
	} else if (x <= 105000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x / y)
    t_1 = x / (1.0d0 + x)
    if (x <= (-330000.0d0)) then
        tmp = t_0
    else if (x <= 2.8d-91) then
        tmp = t_1
    else if (x <= 1.65d-51) then
        tmp = x / (y / x)
    else if (x <= 105000000.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (x / y);
	double t_1 = x / (1.0 + x);
	double tmp;
	if (x <= -330000.0) {
		tmp = t_0;
	} else if (x <= 2.8e-91) {
		tmp = t_1;
	} else if (x <= 1.65e-51) {
		tmp = x / (y / x);
	} else if (x <= 105000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (x / y)
	t_1 = x / (1.0 + x)
	tmp = 0
	if x <= -330000.0:
		tmp = t_0
	elif x <= 2.8e-91:
		tmp = t_1
	elif x <= 1.65e-51:
		tmp = x / (y / x)
	elif x <= 105000000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(x / y))
	t_1 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (x <= -330000.0)
		tmp = t_0;
	elseif (x <= 2.8e-91)
		tmp = t_1;
	elseif (x <= 1.65e-51)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 105000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (x / y);
	t_1 = x / (1.0 + x);
	tmp = 0.0;
	if (x <= -330000.0)
		tmp = t_0;
	elseif (x <= 2.8e-91)
		tmp = t_1;
	elseif (x <= 1.65e-51)
		tmp = x / (y / x);
	elseif (x <= 105000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -330000.0], t$95$0, If[LessEqual[x, 2.8e-91], t$95$1, If[LessEqual[x, 1.65e-51], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 105000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3e5 or 1.05e8 < x

   1. Initial program 77.1%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{y}\right) - \frac{1}{y}} \]

   5. Taylor expanded in y around -inf 99.9%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot x}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

         \[\leadsto 1 + \color{blue}{\left(-\frac{1 + -1 \cdot x}{y}\right)} \]

      2. unsub-neg 99.9%

         \[\leadsto \color{blue}{1 - \frac{1 + -1 \cdot x}{y}} \]

      3. neg-mul-1 99.9%

         \[\leadsto 1 - \frac{1 + \color{blue}{\left(-x\right)}}{y} \]

      4. unsub-neg 99.9%

         \[\leadsto 1 - \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 99.9%

      \[\leadsto \color{blue}{1 - \frac{1 - x}{y}} \]

   8. Taylor expanded in x around inf 99.4%

      \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. neg-mul-1 99.4%

         \[\leadsto 1 - \color{blue}{\left(-\frac{x}{y}\right)} \]

      2. distribute-neg-frac 99.4%

         \[\leadsto 1 - \color{blue}{\frac{-x}{y}} \]

   10. Simplified 99.4%

       \[\leadsto 1 - \color{blue}{\frac{-x}{y}} \]

   if -3.3e5 < x < 2.8e-91 or 1.64999999999999986e-51 < x < 1.05e8

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in y around inf 82.5%

      \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

   if 2.8e-91 < x < 1.64999999999999986e-51

   1. Initial program 99.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]

   2. Taylor expanded in x around inf 94.9%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{x + 1} \]
   3. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{x + 1} \]

      2. associate-*r/ 94.7%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   4. Simplified 94.7%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   5. Taylor expanded in x around 0 94.9%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \]
   6. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \]

      2. associate-*r/ 94.7%

         \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]

   7. Simplified 94.7%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]
   8. Step-by-step derivation

      1. clear-num 94.7%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{x}}} \]

      2. un-div-inv 95.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{x}}} \]

   9. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -330000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 105000000:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 5: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x}\\ \mathbf{if}\;x \leq -13500:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 14500000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ 1.0 x))))
   (if (<= x -13500.0)
     (+ 1.0 (/ (+ x -1.0) y))
     (if (<= x 2.8e-91)
       t_0
       (if (<= x 3.6e-52)
         (/ x (/ y x))
         (if (<= x 14500000.0) t_0 (+ 1.0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -13500.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 2.8e-91) {
		tmp = t_0;
	} else if (x <= 3.6e-52) {
		tmp = x / (y / x);
	} else if (x <= 14500000.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 + x)
    if (x <= (-13500.0d0)) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else if (x <= 2.8d-91) then
        tmp = t_0
    else if (x <= 3.6d-52) then
        tmp = x / (y / x)
    else if (x <= 14500000.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 + (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -13500.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 2.8e-91) {
		tmp = t_0;
	} else if (x <= 3.6e-52) {
		tmp = x / (y / x);
	} else if (x <= 14500000.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (1.0 + x)
	tmp = 0
	if x <= -13500.0:
		tmp = 1.0 + ((x + -1.0) / y)
	elif x <= 2.8e-91:
		tmp = t_0
	elif x <= 3.6e-52:
		tmp = x / (y / x)
	elif x <= 14500000.0:
		tmp = t_0
	else:
		tmp = 1.0 + (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (x <= -13500.0)
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	elseif (x <= 2.8e-91)
		tmp = t_0;
	elseif (x <= 3.6e-52)
		tmp = Float64(x / Float64(y / x));
	elseif (x <= 14500000.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (1.0 + x);
	tmp = 0.0;
	if (x <= -13500.0)
		tmp = 1.0 + ((x + -1.0) / y);
	elseif (x <= 2.8e-91)
		tmp = t_0;
	elseif (x <= 3.6e-52)
		tmp = x / (y / x);
	elseif (x <= 14500000.0)
		tmp = t_0;
	else
		tmp = 1.0 + (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -13500.0], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-91], t$95$0, If[LessEqual[x, 3.6e-52], N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 14500000.0], t$95$0, N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -13500

   1. Initial program 68.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{y}\right) - \frac{1}{y}} \]

   5. Taylor expanded in y around -inf 99.8%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot x}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.8%

         \[\leadsto 1 + \color{blue}{\left(-\frac{1 + -1 \cdot x}{y}\right)} \]

      2. unsub-neg 99.8%

         \[\leadsto \color{blue}{1 - \frac{1 + -1 \cdot x}{y}} \]

      3. neg-mul-1 99.8%

         \[\leadsto 1 - \frac{1 + \color{blue}{\left(-x\right)}}{y} \]

      4. unsub-neg 99.8%

         \[\leadsto 1 - \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \frac{1 - x}{y}} \]

   if -13500 < x < 2.8e-91 or 3.59999999999999988e-52 < x < 1.45e7

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in y around inf 82.5%

      \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

   if 2.8e-91 < x < 3.59999999999999988e-52

   1. Initial program 99.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]

   2. Taylor expanded in x around inf 94.9%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{x + 1} \]
   3. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{x + 1} \]

      2. associate-*r/ 94.7%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   4. Simplified 94.7%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   5. Taylor expanded in x around 0 94.9%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \]
   6. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \]

      2. associate-*r/ 94.7%

         \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]

   7. Simplified 94.7%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]
   8. Step-by-step derivation

      1. clear-num 94.7%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{x}}} \]

      2. un-div-inv 95.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{x}}} \]

   9. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{x}}} \]

   if 1.45e7 < x

   1. Initial program 85.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{y}\right) - \frac{1}{y}} \]

   5. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot x}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto 1 + \color{blue}{\left(-\frac{1 + -1 \cdot x}{y}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{1 - \frac{1 + -1 \cdot x}{y}} \]

      3. neg-mul-1 100.0%

         \[\leadsto 1 - \frac{1 + \color{blue}{\left(-x\right)}}{y} \]

      4. unsub-neg 100.0%

         \[\leadsto 1 - \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{1 - x}{y}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto 1 - \color{blue}{\left(-\frac{x}{y}\right)} \]

      2. distribute-neg-frac 100.0%

         \[\leadsto 1 - \color{blue}{\frac{-x}{y}} \]

   10. Simplified 100.0%

       \[\leadsto 1 - \color{blue}{\frac{-x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -13500:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 14500000:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 6: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 + x}\\ \mathbf{if}\;x \leq -65000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;x \leq 495000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ 1.0 x))))
   (if (<= x -65000.0)
     (+ 1.0 (/ (+ x -1.0) y))
     (if (<= x 2.8e-91)
       t_0
       (if (<= x 5.8e-52)
         (/ 1.0 (/ y (* x x)))
         (if (<= x 495000000.0) t_0 (+ 1.0 (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -65000.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 2.8e-91) {
		tmp = t_0;
	} else if (x <= 5.8e-52) {
		tmp = 1.0 / (y / (x * x));
	} else if (x <= 495000000.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (1.0d0 + x)
    if (x <= (-65000.0d0)) then
        tmp = 1.0d0 + ((x + (-1.0d0)) / y)
    else if (x <= 2.8d-91) then
        tmp = t_0
    else if (x <= 5.8d-52) then
        tmp = 1.0d0 / (y / (x * x))
    else if (x <= 495000000.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 + (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (1.0 + x);
	double tmp;
	if (x <= -65000.0) {
		tmp = 1.0 + ((x + -1.0) / y);
	} else if (x <= 2.8e-91) {
		tmp = t_0;
	} else if (x <= 5.8e-52) {
		tmp = 1.0 / (y / (x * x));
	} else if (x <= 495000000.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (1.0 + x)
	tmp = 0
	if x <= -65000.0:
		tmp = 1.0 + ((x + -1.0) / y)
	elif x <= 2.8e-91:
		tmp = t_0
	elif x <= 5.8e-52:
		tmp = 1.0 / (y / (x * x))
	elif x <= 495000000.0:
		tmp = t_0
	else:
		tmp = 1.0 + (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(1.0 + x))
	tmp = 0.0
	if (x <= -65000.0)
		tmp = Float64(1.0 + Float64(Float64(x + -1.0) / y));
	elseif (x <= 2.8e-91)
		tmp = t_0;
	elseif (x <= 5.8e-52)
		tmp = Float64(1.0 / Float64(y / Float64(x * x)));
	elseif (x <= 495000000.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (1.0 + x);
	tmp = 0.0;
	if (x <= -65000.0)
		tmp = 1.0 + ((x + -1.0) / y);
	elseif (x <= 2.8e-91)
		tmp = t_0;
	elseif (x <= 5.8e-52)
		tmp = 1.0 / (y / (x * x));
	elseif (x <= 495000000.0)
		tmp = t_0;
	else
		tmp = 1.0 + (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -65000.0], N[(1.0 + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-91], t$95$0, If[LessEqual[x, 5.8e-52], N[(1.0 / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 495000000.0], t$95$0, N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -65000

   1. Initial program 68.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{y}\right) - \frac{1}{y}} \]

   5. Taylor expanded in y around -inf 99.8%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot x}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.8%

         \[\leadsto 1 + \color{blue}{\left(-\frac{1 + -1 \cdot x}{y}\right)} \]

      2. unsub-neg 99.8%

         \[\leadsto \color{blue}{1 - \frac{1 + -1 \cdot x}{y}} \]

      3. neg-mul-1 99.8%

         \[\leadsto 1 - \frac{1 + \color{blue}{\left(-x\right)}}{y} \]

      4. unsub-neg 99.8%

         \[\leadsto 1 - \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{1 - \frac{1 - x}{y}} \]

   if -65000 < x < 2.8e-91 or 5.8000000000000003e-52 < x < 4.95e8

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in y around inf 82.5%

      \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

   if 2.8e-91 < x < 5.8000000000000003e-52

   1. Initial program 99.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]

   2. Taylor expanded in x around inf 94.9%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{x + 1} \]
   3. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{x + 1} \]

      2. associate-*r/ 94.7%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   4. Simplified 94.7%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   5. Taylor expanded in x around 0 94.9%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \]
   6. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \]

      2. associate-*r/ 94.7%

         \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]

   7. Simplified 94.7%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 94.9%

         \[\leadsto \color{blue}{\frac{x \cdot x}{y}} \]

      2. clear-num 95.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x \cdot x}}} \]

   9. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x \cdot x}}} \]

   if 4.95e8 < x

   1. Initial program 85.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{y}\right) - \frac{1}{y}} \]

   5. Taylor expanded in y around -inf 100.0%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{1 + -1 \cdot x}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto 1 + \color{blue}{\left(-\frac{1 + -1 \cdot x}{y}\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{1 - \frac{1 + -1 \cdot x}{y}} \]

      3. neg-mul-1 100.0%

         \[\leadsto 1 - \frac{1 + \color{blue}{\left(-x\right)}}{y} \]

      4. unsub-neg 100.0%

         \[\leadsto 1 - \frac{\color{blue}{1 - x}}{y} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{1 - x}{y}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto 1 - \color{blue}{-1 \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto 1 - \color{blue}{\left(-\frac{x}{y}\right)} \]

      2. distribute-neg-frac 100.0%

         \[\leadsto 1 - \color{blue}{\frac{-x}{y}} \]

   10. Simplified 100.0%

       \[\leadsto 1 - \color{blue}{\frac{-x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -65000:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot x}}\\ \mathbf{elif}\;x \leq 495000000:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{y}\\ \end{array} \]

Alternative 7: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-91}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 3100000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ x y)
   (if (<= x 2.8e-91)
     x
     (if (<= x 1.5e-52) (* x (/ x y)) (if (<= x 3100000.0) x (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 2.8e-91) {
		tmp = x;
	} else if (x <= 1.5e-52) {
		tmp = x * (x / y);
	} else if (x <= 3100000.0) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 2.8d-91) then
        tmp = x
    else if (x <= 1.5d-52) then
        tmp = x * (x / y)
    else if (x <= 3100000.0d0) then
        tmp = x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 2.8e-91) {
		tmp = x;
	} else if (x <= 1.5e-52) {
		tmp = x * (x / y);
	} else if (x <= 3100000.0) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 2.8e-91:
		tmp = x
	elif x <= 1.5e-52:
		tmp = x * (x / y)
	elif x <= 3100000.0:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 2.8e-91)
		tmp = x;
	elseif (x <= 1.5e-52)
		tmp = Float64(x * Float64(x / y));
	elseif (x <= 3100000.0)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 2.8e-91)
		tmp = x;
	elseif (x <= 1.5e-52)
		tmp = x * (x / y);
	elseif (x <= 3100000.0)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 2.8e-91], x, If[LessEqual[x, 1.5e-52], N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3100000.0], x, N[(x / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 3.1e6 < x

   1. Initial program 77.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around inf 74.4%

      \[\leadsto \frac{x}{\color{blue}{y}} \]

   if -1 < x < 2.8e-91 or 1.5e-52 < x < 3.1e6

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around 0 79.3%

      \[\leadsto \color{blue}{x} \]

   if 2.8e-91 < x < 1.5e-52

   1. Initial program 99.2%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]

   2. Taylor expanded in x around inf 94.9%

      \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y}}}{x + 1} \]
   3. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y}}{x + 1} \]

      2. associate-*r/ 94.7%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   4. Simplified 94.7%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{x}{y}}}{x + 1} \]

   5. Taylor expanded in x around 0 94.9%

      \[\leadsto \color{blue}{\frac{{x}^{2}}{y}} \]
   6. Step-by-step derivation

      1. unpow2 94.9%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y} \]

      2. associate-*r/ 94.7%

         \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]

   7. Simplified 94.7%

      \[\leadsto \color{blue}{x \cdot \frac{x}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-91}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 3100000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 8: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3100000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ x y) (if (<= x 3100000.0) x (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 3100000.0) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x / y
    else if (x <= 3100000.0d0) then
        tmp = x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = x / y;
	} else if (x <= 3100000.0) {
		tmp = x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = x / y
	elif x <= 3100000.0:
		tmp = x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x / y);
	elseif (x <= 3100000.0)
		tmp = x;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x / y;
	elseif (x <= 3100000.0)
		tmp = x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(x / y), $MachinePrecision], If[LessEqual[x, 3100000.0], x, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 3.1e6 < x

   1. Initial program 77.5%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around inf 74.4%

      \[\leadsto \frac{x}{\color{blue}{y}} \]

   if -1 < x < 3.1e6

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around 0 75.9%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 3100000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 9: 49.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.0) 1.0 (if (<= x 1.0) x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 1.0d0
    else if (x <= 1.0d0) then
        tmp = x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0;
	} else if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = 1.0
	elif x <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 1.0;
	elseif (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], 1.0, If[LessEqual[x, 1.0], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 77.8%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in y around inf 27.2%

      \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

   5. Taylor expanded in x around inf 25.2%

      \[\leadsto \color{blue}{1} \]

   if -1 < x < 1

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

   4. Taylor expanded in x around 0 76.9%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 14.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 89.0%

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
2. Step-by-step derivation

   1. associate-/l* 99.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]

4. Taylor expanded in y around inf 53.2%

   \[\leadsto \frac{x}{\color{blue}{1 + x}} \]

5. Taylor expanded in x around inf 14.2%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 14.2%

   \[\leadsto 1 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0))))

C

double code(double x, double y) {
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / 1.0d0) * (((x / y) + 1.0d0) / (x + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
}

Python

def code(x, y):
	return (x / 1.0) * (((x / y) + 1.0) / (x + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x / 1.0) * Float64(Float64(Float64(x / y) + 1.0) / Float64(x + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x / 1.0) * (((x / y) + 1.0) / (x + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x / 1.0), $MachinePrecision] * N[(N[(N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023199 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))